Orbital eccentricities in exo-planetary systems discovered
thus far are often surprisingly large and have proven to be
a major puzzle in understanding these systems. In
multiple-planet systems, there is also a tendency for
near-alignment of the arguments of periapse of two planets,
and secular apsidal resonance is strongly indicated in one
case, Upsilon Andromedae. Apsidal resonance is the
phenomenon of phase-locking of the apsidal longitudes of two
orbits, such that the two planets have a common average rate
of apsidal precession and the angular difference of their
apsidal longitudes librates around 0. We describe a
dynamical mechanism for establishing apsidal resonance in a
pair of planets that are initially on nearly circular
orbits. We show that in such a system, a perturbation that
imparts a finite eccentricity to one planet's orbit causes
the other planet's orbit to become eccentric as well, and
also leads to a libration of their relative apsidal
longitudes for a wide range of initial conditions. Such a
mechanism may explain orbital eccentricities and apsidal
alignment in some exo-planetary systems. The eccentricity
perturbation could be caused by the ejection of a planet
from these systems or by torques from a primordial gas disk.
The amplitude of secular variations provides a constraint on
the dynamical history of such systems.